A math definition of orbifold Euler characteristic appears in On the Euler Number of an Orbifold. If there is a group acting on a manifold $X / G$ we could try to write
\[ \chi(X/G) = \frac{1}{|G|} \sum_{g \in G} \chi(X^g) \]
where $X^g$ is the fixed-point set of $g \in G$. Instead it seems to be better to sum over conjugacy classes $[g]$ of $G$.
\[ \chi(X,G) = \sum_{[g]} \chi(X/C(g)) \]
where $C(g) = \{ h: hgh^{-1}=g\}$ is the centralizer of $g \in G$. For any group action $|[g]|\cdot |C(g)| = G$.
This definition was motivated by some physicists in the last 1980's and by the mid 90's this idea was extended to homology. See A Strong Coupling Test of S-Duality
\[ H^(X/G) = \bigoplus_g H^(X^g)^{C(g)} \]
The direct sum of the centralizer-invariant part $[\cdot] ^{C(g)}$ of the cohomology classes $H^*(\cdot)$ of fixed point sets $X^g$. As $g$ runs over the conjugacy classes of $G$.
This was used to find the Euler characteristic of the Hilbert scheme of points on a non-singular space (originally found by Lothar Göttsche)
\[ \sum_{n=0}^\infty q^n \chi(X^{[n]}) = \frac{1}{\prod_{i=1}^\infty (1-q^n)^{\chi(X)}}\]
So there's connection to the Dedekind η-function. This generating function might be different if $X$ itself is singular.
